Thermodynamics of Fluid Polyamorphism

Fluid polyamorphism is the existence of different condensed amorphous states in a single-component fluid. It is either found or predicted, usually at extreme conditions, for a broad group of very different substances, including helium, carbon, silicon, phosphorous, sulfur, tellurium, cerium, hydrogen, and tin tetraiodide. This phenomenon is also hypothesized for metastable and deeply supercooled water, presumably located a few degrees below the experimental limit of homogeneous ice formation. We present a generic phenomenological approach to describe polyamorphism in a single-component fluid, which is completely independent of the molecular origin of the phenomenon. We show that fluid polyamorphism may occur either in the presence or in the absence of fluid phase separation depending on the symmetry of the order parameter. In the latter case, it is associated with a second-order transition, such as in liquid helium or liquid sulfur. To specify the phenomenology, we consider a fluid with thermodynamic equilibrium between two distinct interconvertible states or molecular structures. A fundamental signature of this concept is the identification of the equilibrium fraction of molecules involved in each of these alternative states. However, the existence of the alternative structures may result in polyamorphic fluid phase separation only if mixing of these structures is not ideal. The two-state thermodynamics unifies all the debated scenarios of fluid polyamorphism in different areas of condensed-matter physics, with or without phase separation, and even goes beyond the phenomenon of polyamorphism by generically describing the anomalous properties of fluids exhibiting interconversion of alternative molecular states.

Popular Summary

For most pure materials (at a given temperature and pressure), molecules are arranged in a single noncrystalline or amorphous structure that is liquid, gas, or solid. Conventional theory adequately predicts the thermodynamic properties of such materials, which are well understood. In contrast to this simple picture, there is increasing evidence that many pure fluids exist in several alternative structures, resulting in the phenomenon known as “fluid polyamorphism.” It has been hypothesized that even water may exhibit this behavior at very low temperatures. This work contributes to our understanding of this phenomenon by developing a unified theoretical framework that is capable of making robust predictions for the thermodynamic properties of different polyamorphic fluids.

We use the theory of phase transitions and the concept of two competing interconvertible amorphous structures to develop a generic phenomenological approach, which is independent of the underlying molecular origin of the phenomenon and provides new physical insights. It succeeds in unifying all, apparently unrelated, cases of fluid polyamorphism with and without phase separation, from the liquid-liquid transition in supercooled water and silicon to superfluid helium and polymerized sulfur. Our approach generically describes the phase behavior and thermodynamic anomalies typically observed in polyamorphic materials, including liquid-vapor and liquid-liquid transitions, as well as stretched metastable liquid states under negative pressures.

Our results mark a paradigm shift that significantly broadens fluid polyamorphism from its original narrow scope to a cross-disciplinary field that addresses a wide class of systems and phenomena with interconversion of alternative molecular or supramolecular states.

Figure 1

Phase diagram for a polyamorphic lattice gas: $\widehat{T}=T/T_{c1}$ and $\widehat{p}=p/p_{c1}$, $\rho$ is the density. (a) Pressure-temperature diagram. The blue curves are either vapor-liquid or liquid-liquid transitions; CP1 and CP2 are the vapor-liquid and liquid-liquid critical points assigned (as an example) to be at the same isobar. The red dotted curves are the liquid and vapor branches of liquid-vapor spinodal and the blue dotted line is the Widom line. (b) Temperature-density diagram. The thick blue and thick red curves are the vapor-liquid and liquid-liquid coexistence, respectively. The multicolor curves are selected isobars.

Figure 2

The dimensionless isothermal compressibility ${\widehat{\kappa }}_T={\rho }^{-1}(\partial \rho /\partial \widehat{p})$ along three selected isobars: above the liquid-liquid critical pressure, assigned equal to the vapor-liquid critical pressure (green line), at the critical pressure (red line), and below the critical pressure (blue line). ${\widehat{T}}_{c1}=1$ and ${\widehat{T}}_{c2}=T_{c2}/T_{c1}$ are the vapor-liquid and liquid-liquid critical temperatures, ${\widehat{T}}_{\mathrm{LL}}=T_{\mathrm{LL}}/T_{c1}$ is the temperature of the liquid-liquid transition at the selected isobar. $T_{\text{SP}}$ is the temperature of the vapor-liquid spinodal.

Figure 3

Parametrized phase diagram for polyamorphic lattice gas in terms of nonideality of mixing of the alternative states $A$ and $B$, ${\widehat{T}}_{c2}=\omega /2k{T}_{c1}$, and the energy difference of $A$ and $B$, $\widehat{\lambda }=\lambda /k{T}_{c1}$. The volume difference of states $A$ and $B$ is taken constant. The singularity-free scenario corresponds to ${\widehat{T}}_{c2}=0$. The critical-point-free scenario is favored by stronger nonideality and smaller energy difference.

Figure 4

Evolution of the pattern of the extrema loci upon tuning the location of the liquid-liquid critical point. Black is the density maximum or minimum; red is the isothermal compressibility maximum or minimum along isobars; green is the isobaric heat capacity maximum or minimum along isotherms; dotted green shows additional (shallow) extrema of the heat capacity unrelated to the liquid-liquid transition; dotted red are two branched of the liquid-vapor spinodal; blue dashed is the Widom line; red dots are the vapor-liquid (CP1) and the liquid-liquid (CP2) critical points. (a) A “singularity-free” scenario—the critical point is at zero temperature, thus it is not labeled as CP2; extrapolations of the extrema loci to zero temperature are shown as dashed lines. (b) A “regular” scenario—the critical point CP2 is at a positive pressure. (c) The critical point coincides with the absolute stability limit of the liquid state. (d) A “critical point-free” scenario—the “virtual” critical point CP2 is located in the unstable region.

Figure 5

A singularity (“bird’s beak”) in the liquid-liquid coexistence curve if the critical point coincides with the liquid-vapor spinodal. (a) Pressure-density, (b) Temperature-density. Thick blue curve is the vapor-liquid coexistence; thick red is the liquid-liquid coexistence; CP1 and CP2 are the vapor-liquid and liquid-liquid critical points, respectively; dotted blue is the liquid-vapor spinodal; multicolor curves are selected isotherms (a) and selected isobars (b).

Figure 6

Fraction of polymerized monomers as a function of temperature for different degrees of polymerization. Green curve corresponds to $N=10$, blue curve to $N=30$, and red curve to $N\to \infty$. The polymerization in the limit $N\to \infty$ is equivalent to a continuous (second-order) phase transition in zero field.

Figure 7

Generalized phase diagram of a fluid exhibiting equilibrium polymerization into an infinite chain. Blue area is the polymeric phase; thick black line and curve are first-order phase transitions (coexistence between monomer and polymer phases and between vapor and liquid, respectively); $CP$ is the vapor-liquid critical point; the theta point is equivalent to a tricritical point which separates the second-order and first-order phase transitions; TRP is the triple point (monomers, polymer-enriched phase, and vapor coexist). Dotted red curve represents the absolute stability limit of liquid with respect to vapor. Blue dashed line is continuation of the liquid-liquid transition line into the metastable region.